We solve the Optimal Power Flow (OPF) problem through dual relaxation. A sufficient condition under which the dual solution is exact is satisfied by standard IEEE benchmark systems with 14, 30, 57, 118 and 300 buses, when a small resistance is added to transformers originally assumed to have zero resistance. In integrating renewable energy (such as wind power) into the electric grid, difficulties due to source-intermittency, demand-fluctuations, and large distances between generation sites and users can be overcome with a
transmission network with large-scale storage. For the special case of a single generator and a single load, we prove that the optimal generation schedule for the finite-horizon OPF problem with storage initially generates more than the demand, in order to charge up the battery, and then generates less than the demand and uses the battery to supplement generation in the final stages. (Joint work with Mani Chandy, Javad Lavaei, Ufuk Topcu, Mumu Xu).